TITLE:
The Tarski Problems and Their Solutions
AUTHORS:
Benjamin Fine, Anthony Gaglione, Gerhard Rosenberger, Dennis Spellman
KEYWORDS:
Non-Abelian Free Group, Elementary Theory, Tarski Problems, Elementary Free Groups, Algebraic Geometry over Groups
JOURNAL NAME:
Advances in Pure Mathematics,
Vol.5 No.4,
March
31,
2015
ABSTRACT: Around 1945, Alfred Tarski proposed several questions concerning the
elementary theory of non-abelian free groups. These remained open for 60 years
until they were proved by O. Kharlampovich and A. Myasnikov and independently
by Z. Sela. The proofs, by both sets of authors, were monumental and involved
the development of several new areas of infinite group theory. In this paper we
explain precisely the Tarski problems and what has been actually proved. We
then discuss the history of the solution as
well as the components of the proof. We then provide the basic strategy
for the proof. We finish this paper with a brief discussion of elementary free
groups.